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February  2020, 19(2): 771-783. doi: 10.3934/cpaa.2020036

Liouville theorems for an integral equation of Choquard type

Phuong Le 1,2,

1. 

Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
2. 

Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

Received  November 2018 Revised  July 2019 Published  October 2019

We establish sharp Liouville theorems for the integral equation
$ u(x) = \int_{\mathbb{R}^n} \frac{u^{p-1}(y)}{|x-y|^{n-\alpha}} \int_{\mathbb{R}^n} \frac{u^p(z)}{|y-z|^{n-\beta}} dz dy, \quad x\in\mathbb{R}^n, $
where
$ 0<\alpha, \beta<n $
 and
$ p>1 $
. Our results hold true for positive solutions under appropriate assumptions on
$ p $
 and integrability of the solutions. As a consequence, we derive a Liouville theorem for positive
$ H^{\frac{\alpha}{2}}(\mathbb{R}^n) $
 solutions of the higher fractional order Choquard type equation
$ (-\Delta)^{\frac{\alpha}{2}} u = \left(\frac{1}{|x|^{n-\beta}} * u^p\right) u^{p-1} \quad\text{ in } \mathbb{R}^n. $
Keywords: Choquard equation, integral equation, fractional Laplacian, Liouville theorem, nonexistence of positive solutions.
Mathematics Subject Classification: Primary: 35R11, 35J91; Secondary: 45G10, 35B53.
Citation: Phuong Le. Liouville theorems for an integral equation of Choquard type. Communications on Pure & Applied Analysis, 2020, 19 (2) : 771-783. doi: 10.3934/cpaa.2020036
References:
[1]

D. Applebaum, Lévy processes - from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.   Google Scholar

[2]

P. d'Avenia, G. Siciliano and M. Squassina, On fractional Choquard equations, Math. Models Methods Appl. Sci., 25 (2015), 1447–1476. doi: 10.1142/S0218202515500384.  Google Scholar

[3]

P. Belchior, H. Bueno, O. H. Miyagaki and G. A. Pereira, Remarks about a fractional Choquard equation: Ground state, regularity and polynomial decay, Nonlinear Anal., 164 (2017), 38-53.  doi: 10.1016/j.na.2017.08.005.  Google Scholar

[4]

J. Bertoin, Lévy Processes, Cambridge University Press, 1996.  Google Scholar

[5]

J. P. Bouchard and A. Georges, Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293.  doi: 10.1016/0370-1573(90)90099-N.  Google Scholar

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L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Annals of Math., 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903.  Google Scholar

[7]

D. Cao and W. Dai, Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 979-994.  doi: 10.1017/prm.2018.67.  Google Scholar

[8]

G. Caristi, L. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67.  doi: 10.1007/s00032-008-0090-3.  Google Scholar

[9]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar

[10]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445.  Google Scholar

[11]

W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Math. Sci., 29B (2009), 949-960.  doi: 10.1016/S0252-9602(09)60079-5.  Google Scholar

[12]

P. Constantin, Euler Equations, Navier-Stokes Equations and Turbulence, Mathematical Foundation of Turbulent Viscous Flows, Vol. 1871 of lecture Notes in Math. 1–43, Springer, Berlin, 2006. doi: 10.1007/11545989_1.  Google Scholar

[13]

W. Dai, Y. Fang, J. Huang, Y. Qin and B. Wang, Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete Contin. Dyn. Syst., 39 (2019), 1389-1403.  doi: 10.3934/dcds.2018117.  Google Scholar

[14]

P. Le, Symmetry and classification of solutions to an integral equation of Choquard type, submitted for publication. Google Scholar

[15]

P. Le, Liouville theorem and classification of positive solutions for a fractional Choquard type equation, Nonlinear Anal., 185 (2019), 123-141. doi: 10.1016/j.na.2019.03.006.  Google Scholar

[16]

Y. Lei, Liouville theorems and classification results for a nonlocal Schrödinger equation, Discrete Contin. Dyn. Syst., 38 (2018), 5351-5377.  doi: 10.3934/dcds.2018236.  Google Scholar

[17]

Y. Lei, On the regularity of positive solutions of a class of Choquard type equations, Math. Z., 273 (2013), 883-905.  doi: 10.1007/s00209-012-1036-6.  Google Scholar

[18]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar

[19]

E. Lieb, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977) 185–194.  Google Scholar

[20]

S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., 71 (2009), 1796-1806.  doi: 10.1016/j.na.2009.01.014.  Google Scholar

[21]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.  Google Scholar

[22]

I. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.  doi: 10.1088/0264-9381/15/9/019.  Google Scholar

[23]

V. Moroz and J. V. Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1.  Google Scholar

[24]

V. Moroz and J. V. Schaftingen, Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains}, J. Differential Equations, 254 (2013), 3089-3145.  doi: 10.1016/j.jde.2012.12.019.  Google Scholar

[25]

S. Pekar, Untersuchungen über die Elekronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. Google Scholar

[26]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, New Jersey, 1970.  Google Scholar

[27]

V. Tarasov and G. Zaslasvky, Fractional dynamics of systems with long-range interaction, Comm. Nonl. Sci. Numer. Simul., 11 (2006), 885-889.  doi: 10.1016/j.cnsns.2006.03.005.  Google Scholar

[28]

D. Xu and Y. Lei, Classification of positive solutions for a static Schrödinger-Maxwell equation with fractional Laplacian, Applied Math. Letters, 43 (2015), 85-89.  doi: 10.1016/j.aml.2014.12.007.  Google Scholar

[29]

W. Zhang and X. Wu, Nodal solutions for a fractional Choquard equation, J. Math. Anal. Appl., 464 (2018), 1167-1183.  doi: 10.1016/j.jmaa.2018.04.048.  Google Scholar

show all references

References:
[1]

D. Applebaum, Lévy processes - from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.   Google Scholar

[2]

P. d'Avenia, G. Siciliano and M. Squassina, On fractional Choquard equations, Math. Models Methods Appl. Sci., 25 (2015), 1447–1476. doi: 10.1142/S0218202515500384.  Google Scholar

[3]

P. Belchior, H. Bueno, O. H. Miyagaki and G. A. Pereira, Remarks about a fractional Choquard equation: Ground state, regularity and polynomial decay, Nonlinear Anal., 164 (2017), 38-53.  doi: 10.1016/j.na.2017.08.005.  Google Scholar

[4]

J. Bertoin, Lévy Processes, Cambridge University Press, 1996.  Google Scholar

[5]

J. P. Bouchard and A. Georges, Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293.  doi: 10.1016/0370-1573(90)90099-N.  Google Scholar

[6]

L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Annals of Math., 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903.  Google Scholar

[7]

D. Cao and W. Dai, Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 979-994.  doi: 10.1017/prm.2018.67.  Google Scholar

[8]

G. Caristi, L. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67.  doi: 10.1007/s00032-008-0090-3.  Google Scholar

[9]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar

[10]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445.  Google Scholar

[11]

W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Math. Sci., 29B (2009), 949-960.  doi: 10.1016/S0252-9602(09)60079-5.  Google Scholar

[12]

P. Constantin, Euler Equations, Navier-Stokes Equations and Turbulence, Mathematical Foundation of Turbulent Viscous Flows, Vol. 1871 of lecture Notes in Math. 1–43, Springer, Berlin, 2006. doi: 10.1007/11545989_1.  Google Scholar

[13]

W. Dai, Y. Fang, J. Huang, Y. Qin and B. Wang, Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete Contin. Dyn. Syst., 39 (2019), 1389-1403.  doi: 10.3934/dcds.2018117.  Google Scholar

[14]

P. Le, Symmetry and classification of solutions to an integral equation of Choquard type, submitted for publication. Google Scholar

[15]

P. Le, Liouville theorem and classification of positive solutions for a fractional Choquard type equation, Nonlinear Anal., 185 (2019), 123-141. doi: 10.1016/j.na.2019.03.006.  Google Scholar

[16]

Y. Lei, Liouville theorems and classification results for a nonlocal Schrödinger equation, Discrete Contin. Dyn. Syst., 38 (2018), 5351-5377.  doi: 10.3934/dcds.2018236.  Google Scholar

[17]

Y. Lei, On the regularity of positive solutions of a class of Choquard type equations, Math. Z., 273 (2013), 883-905.  doi: 10.1007/s00209-012-1036-6.  Google Scholar

[18]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar

[19]

E. Lieb, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977) 185–194.  Google Scholar

[20]

S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., 71 (2009), 1796-1806.  doi: 10.1016/j.na.2009.01.014.  Google Scholar

[21]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.  Google Scholar

[22]

I. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.  doi: 10.1088/0264-9381/15/9/019.  Google Scholar

[23]

V. Moroz and J. V. Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1.  Google Scholar

[24]

V. Moroz and J. V. Schaftingen, Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains}, J. Differential Equations, 254 (2013), 3089-3145.  doi: 10.1016/j.jde.2012.12.019.  Google Scholar

[25]

S. Pekar, Untersuchungen über die Elekronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. Google Scholar

[26]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, New Jersey, 1970.  Google Scholar

[27]

V. Tarasov and G. Zaslasvky, Fractional dynamics of systems with long-range interaction, Comm. Nonl. Sci. Numer. Simul., 11 (2006), 885-889.  doi: 10.1016/j.cnsns.2006.03.005.  Google Scholar

[28]

D. Xu and Y. Lei, Classification of positive solutions for a static Schrödinger-Maxwell equation with fractional Laplacian, Applied Math. Letters, 43 (2015), 85-89.  doi: 10.1016/j.aml.2014.12.007.  Google Scholar

[29]

W. Zhang and X. Wu, Nodal solutions for a fractional Choquard equation, J. Math. Anal. Appl., 464 (2018), 1167-1183.  doi: 10.1016/j.jmaa.2018.04.048.  Google Scholar

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